Topology

Title: Counting associatives and Seiberg-Witten equations

Date: 10/02/2017

Time: 4:10 PM - 5:30 PM

Place: C304 Wells Hall

Speaker: Thomas Walpuski, MSU

There is a natural functional on the space of orientation 3-dimensional submanifolds in a G2-manifold. Its critical points are associative submanifolds, a special class of volume-minimizing submanifolds which obey an elliptic deformation theory. Given this, it is a natural question whether one can count associative submanifolds in order to construct an enumerative invariant for G2–manifolds. I will explain several geometric scenarios, which prohibit a naive count of such submanifolds cannot possible be invariant. I will then go on to discuss how (generalized) Seiberg-Witten equations might help cure these problems.

Mathematical Physics and Gauge Theory

The notion of “broken symmetry” is central in gauge theories. For BPS monopoles, symmetry breaking can be defined in terms of the eigenvalues of the Higgs-field at infinity. The symmetry breaking is maximal if the eigenvalues are distinct. Monopoles with maximal symmetry breaking have been studied extensively by both mathematicians and physicists for decades now, but little is known about the general case.
In this talk, I will show how to produce monopoles with arbitrary symmetry breaking using the Nahm transform, and I will also outline the construction of its inverse. The inversion heavily uses first order elliptic PDE's on non-compact spaces, more concretely, the theory of Callias-type operators in 3D.
This is a joint project with Benoit Charbonneau.

Combinatorics and Graph Theory

Title: Follow the Rules!

Date: 10/03/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Robert Davis, MSU

Schur functions are among the most useful bases for symmetric functions, but they come at the cost of making certain computations much less obvious than when done in other bases. In particular, how can we determine the coefficients of a product of Schur functions in the basis of Schur functions? There are many equivalent ways to computing these numbers; this talk will discuss jeu de taqin, which is an equivalence among skew tableaux, and apply it to obtain a formulation of the Littlewood-Richardson rule, which answers our question. If time allows, we will discuss other important formulations of this rule.

Colloquium

Title: The Development of Shocks in Compressible Fluids

Date: 10/04/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Demetrios Christodoulou, ETH Zurich

The lecture shall trace the history of the theoretical study of the formation and evolution of shocks in compressible fluids, starting with the fundamental work of Riemann, the first work on nonlinear hyperbolic partial differential equations. Riemann considered the case of plane symmetry where the problem reduces to 1 spatial dimension. One milestone in the development of the theory was the work of Sideris who gave the first general proof of the finite time breakdown of smooth solutions in 3 spatial dimensions. Another milestone was the work of Majda who first addressed the problem of the local in time continuation of a shock front as a nonlinear free boundary problem for a nonlinear hyperbolic system of partial differential equations. I shall then discuss my own work, which uses differential geometric methods and resolves the resulting singularities giving a complete description in terms of smooth functions.
My first work studies the maximal smooth development of given smooth initial data, the boundary of the domain of this development, and the behavior of the solution at this boundary. The boundary contains certain singular hypersurfaces which originate from certain singular surfaces. The singular surfaces do occur in nature, but not the singular hypersurfaces. My second work studies the physical evolution beyond the singular surfaces by solving a nonlinear free boundary problem with singular initial conditions associated to each of the singular surfaces. From each singular surface a shock hypersurface issues which appears as the corresponding free boundary.

Seminar in Cluster algebras

Title: Cluster algebras of geometric type I

Date: 10/05/2017

Time: 10:00 AM - 10:50 AM

Place: C304 Wells Hall

Speaker: Leonid Chekov, Michigan State University

I will describe how cluster algebras arise in hyperbolic geometry of Riemann surfaces \Sigma_{g,s} with s>0 holes with the identification of cluster variables with Penner's lambda lengths, X-variables with shear coordinates, and terms of exchange matrices --- with coefficients of Poisson relations between the shear coordinates. I also describe sets of geodesic functions and their algebras induced by semiclassical/quantum relations for the shear coordinates.

Mathematical Physics and Gauge Theory

Title: Reading Seminar on Topological Insulators

Date: 10/05/2017

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

Speaker: Matthew Cha

In the study of quantum phases, the concept of topological invariant has emerged as a new paradigm beyond that of Landau theory. The relevance of topology for the classification of phases has been known since the discovery of the quantum hall effect. However, recent theoretical and experimental discoveries of new topological insulators has led to a renewed interest. The purpose of this reading group is to explore both recent and classical results for topological insulators including but not limited to (1) bulk-boundary correspondence (2) K-theoretic classification of topological insulators (3) topological invariants in the presence of disorder (4) quantization of Hall conductance in interacting systems.

Geometry and Topology

Title: Wall-crossing for the Seiberg-Witten equation with two spinors

Date: 10/05/2017

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Speaker: Thomas Walpuski, MSU

Unlike for the classical Seiberg-Witten equation, compactness fails for the Seiberg-Witten equation with multiple spinors. This non-compactness is caused by Fueter sections with values in the moduli space of charge 1 SU(n) ASD instantons. In the simplest case, n = 2, those are Z/2 harmonic spinors. In this talk I will explain in more detail what the preceding sentences mean and then discuss the wall-crossing caused by the appearance of (non-singular) Z/2 harmonic spinors. Time permitting, I will discuss how our wall-crossing formulae can be used to prove the existence of singular Fueter sections.
This is joint work with Aleksander Doan.

Colloquium

Title: Nonlinear stability of sources

Date: 10/05/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Björn Sandstede, Brown University

Defects are interfaces that mediate between two wave trains with possibly different wave numbers. Of particular interest in applications are sources for which the group velocities of the wave trains to either side of the defect point away from the interface. While sources are ubiquitous in experiments and can be found easily in numerical simulations of appropriate models, their stability analysis still presents many challenges. One difficulty is that sources are not travelling waves but are time-periodic in an appropriate moving coordinate frame. A second difficulty is that perturbations are transported towards infinity, which makes it difficult to apply various commonly used approaches. In this talk, I will discuss nonlinear stability results for sources in general reaction-diffusion system and outline a proof that utilizes pointwise estimates.